scholarly journals Viscosity approximation method for solving variational inequality problem in real Banach spaces

2021 ◽  
pp. 1-20
Author(s):  
Godwin Ugwunnadi
Keyword(s):  
Variational Inequality ◽  
Banach Spaces ◽  
Approximation Method ◽  
Variational Inequality Problem ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Inequality Problem ◽  
Mild Conditions ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces

In this paper, we study the implicit and inertial-type viscosity approximation method for approximating a solution to the hierarchical variational inequality problem. Under some mild conditions on the parameters, we prove that the sequence generated by the proposed methods converges strongly to a solution of the above-mentioned problem in $q$-uniformly smooth Banach spaces. The results obtained in this paper generalize and improve many recent results in this direction.

scholarly journals Krasnoselskii–Mann Viscosity Approximation Method for Nonexpansive Mappings

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1153
Author(s):  
Najla Altwaijry ◽  
Tahani Aldhaban ◽  
Souhail Chebbi ◽  
Hong-Kun Xu
Keyword(s):  
Approximation Method ◽  
Geometric Property ◽  
Nonexpansive Mappings ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Mann Iteration ◽  
Underlying Space ◽  
Uniformly Smooth ◽  
Regularization Parameters ◽  
Uniformly Smooth Banach Spaces

We show that the viscosity approximation method coupled with the Krasnoselskii–Mann iteration generates a sequence that strongly converges to a fixed point of a given nonexpansive mapping in the setting of uniformly smooth Banach spaces. Our result shows that the geometric property (i.e., uniform smoothness) of the underlying space plays a role in relaxing the conditions on the choice of regularization parameters and step sizes in iterative methods.

scholarly journals Common Attractive Points of Generalized Hybrid Multi-Valued Mappings and Applications

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1307
Author(s):  
Lili Chen ◽  
Ni Yang ◽  
Jing Zhou
Keyword(s):  
Variational Inequality ◽  
Weak Convergence ◽  
Strong Convergence ◽  
Approximation Method ◽  
Variational Inequality Problem ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Banach Limits ◽  
Weak Convergence Theorem ◽  
The Common

In this paper, we first propose the concepts of (ζ,η,λ,π)-generalized hybrid multi-valued mappings, the set of all the common attractive points (CAf,g) and the set of all the common strongly attractive points (CsAf,g), respectively for the multi-valued mappings f and g in a CAT(0) space. Moreover, we give some elementary properties in regard to the sets Af, Ff and CAf,g for the multi-valued mappings f and g in a complete CAT(0) space. Furthermore, we present a weak convergence theorem of common attractive points for two (ζ,η,λ,π)-generalized hybrid multi-valued mappings in the above space by virtue of Banach limits technique and Ishikawa iteration respectively. Finally, we prove strong convergence of a new viscosity approximation method for two (ζ,η,λ,π)-generalized hybrid multi-valued mappings in CAT(0) spaces, which also solves a kind of variational inequality problem.

scholarly journals System of Variational Inclusions and Fixed Points of Pseudocontractive Mappings in Banach Spaces

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 5
Author(s):  
Lu-Chuan Ceng ◽  
Mihai Postolache ◽  
Xiaolong Qin ◽  
Yonghong Yao
Keyword(s):  
Banach Spaces ◽  
Infinite Family ◽  
General System ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Variational Inclusions ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Mild Conditions ◽  
Pseudocontractive Mappings

The purpose of this paper is to solve the general system of variational inclusions (GSVI) with hierarchical variational inequality (HVI) constraint, for an infinite family of continuous pseudocontractive mappings in Banach spaces. By utilizing the equivalence between the GSVI and the fixed point problem, we construct an implicit multiple-viscosity approximation method for solving the GSVI. Under very mild conditions, we prove the strong convergence of the proposed method to a solution of the GSVI with the HVI constraint, for infinitely many pseudocontractions.

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